I bought a hyacinth plant which bloomed and then the flowers died. So once the flowers have died but the greenery still looks fine, is there anything I can do to coax more flowers out of the plant?
Spring bulbs like hyacinths only bloom once a year. Now that this year's blooms have died back, keep the plant in a sunny spot and make sure it doesn't get too dry. Feed it now and then with a weak fertilizer solution when you water. In other words, treat it as a houseplant until you can plant it outside. When the leaves start to yellow and die back this spring or summer, that means the bulb is going dormant for the rest of the year. Once that happens you should plant it in one of your flower beds somewhere in the fall, so it can settle in over the winter. It will send up leaves again in the spring, but may not flower again next year. (Flowering a bulb in a pot generally weakens it somewhat.) If you care for it well next year, as you do other outside flowering plants, it should definitely bloom the next year.
One thing to remember with spring bulbs - never cut a bulb's leaves back until they start to yellow. The leaves are what energizes the bulb so that it has enough oomph to flower again the next season and if you cut them back too soon, the bulb probably won't flower well the next year, if at all, and may eventually weaken enough to die.
It is potted, yes? It will not bloom again this year. Bulbs generally build up enough energy to put out a single flowering. Once that is over, you need to wait for next year.
If your bulb was forced, it may not bloom next year, either. Forcing takes a lot out of the plant, and they need time to recover. Many people throw them away after they are done. I've always thought that is a bit of a shame. I plant them out in my garden. They never seem to look quite as showy as they did the first time, but they are still pretty and brighten up the garden.
Do be sure to let the plant die back on its own if you want to replant it. Now is the time when it is storing the energy for next year's blooms. If you cut off the green foliage, it won't get a chance.